| Summary: | We formulate rigorous criteria for selecting diagrams to be taken into account in the analysis of potential polyquark (tetra or penta) poles in QCD. The central point of these criteria is the requirement that the Feynman diagrams for the relevant Green functions contain four-or five-quark intermediate states and the corresponding cuts. It is shown that some diagrams which “visually”7 seem to contain the four-quark cuts, turn out to be free of these singularities and therefore should not be taken into account when calculating the tetraquark properties. We then consider large-Nc QCD which in many cases provide qualitatively correct picture of hadron properties and discuss in detail the tetraquark states. For the “direct” and the “recombination” four-point Green functions, which may potentially contain the tetraquark poles, we formulate large-Nc consistency conditions which strongly restrict the behaviour of the tetraquark-to-ordinary meson transition amplitudes. In the end, these conditions allow us to obtain constraints on width of the potential tetraquark states at large Nc. We report that both flavor-exotic and cryptoexotic (i.e., flavor-nonexotic) tetraquarks, if the corresponding poles exist, have a width of order
O
1
/
N
c
2
, i.e. they should be parametrically narrower compared to the ordinary
q
¯
q
mesons with a width of order O(1/Nc). Moreover, for flavor-exotic states, the large-Nc consistency conditions require two narrow flavor-exotic states, each of these states coupling dominantly to one specific meson-meson channel.
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